Sharedwww / Tables / ants.texOpen in CoCalc
Author: William A. Stein
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27\begin{document}
28
29\title{Component Groups of Quotients of $J_0(N)$}
30\titlerunning{Component Groups}
31\author{David Kohel\inst{1} \and William A. Stein\inst{2}}
32\authorrunning{Kohel \and Stein}
33\tocauthor{David Kohel (University of Sydney),
34William A. Stein (University of California at Berkeley)}
35%
36\institute{University of Sydney\\
37\email{kohel@maths.usyd.edu.au}\\
38\texttt{http://www.maths.usyd.edu.au:8000/u/kohel/}
39\and
40University of California at Berkeley,\\
41\email{was@math.berkeley.edu}\\
42\texttt{http://shimura.math.berkeley.edu/\~{}was}
43}
44
45\maketitle
46\begin{abstract}
47Let~$f$ be a newform of weight~$2$ on $\Gamma_0(N)$, and let~$A_f$
48be the corresponding optimal abelian variety quotient of $J_0(N)$.
49We describe an algorithm to compute the order of the component group
50of $A_f$ at primes~$p$ that exactly divide~$N$.  We give a table of
51orders of component groups for all~$f$ of level $N\leq 127$ and five
52examples in which the component group is very large, as predicted
53by the Birch and Swinnerton-Dyer conjecture.
54\end{abstract}
55
56\section{Introduction}
57
58Let $X_0(N)$ be the Riemann surface obtained by compactifying the
59quotient of the upper half-plane by the action of $\Gamma_0(N)$.
60Then $X_0(N)$ has a canonical structure of algebraic curve
61over~$\Q$; denote its Jacobian by $J_0(N)$.  It is equipped with
62an action of a commutative ring $\T=\Z[\ldots T_n \ldots]$ of
63Hecke operators.  For more details on modular curves, Hecke operators,
64and modular forms see, e.g.,~\cite{diamond-im}.
65
66Now suppose that~$f=\sum_{n=1}^{\infty} a_n q^n$ is a modular newform
67of weight~$2$ for the congruence subgroup $\Gamma_0(N)$.
68The Hecke operators also act on~$f$ by $T_n(f) = a_n f$.
69The eigenvalues $a_n$ generate an order $R_f = \Z[\ldots a_n \ldots]$
70in a number field $K_f$.
71The kernel $I_f$ of the map $\T \rta R_f$ sending $T_n$ to~$a_n$
72is a prime ideal.
73Following Shimura~\cite{shimura:factors}, we associate to~$f$ the
74quotient $A_f = J_0(N)/I_f J_0(N)$ of
75$J_0(N)$.  Then $A_f$ is an abelian variety over~$\Q$ of dimension
76$[K_f:\Q]$, with bad reduction exactly at the primes dividing~$N$.
77
78One-dimensional quotients of $J_0(N)$ have been intensely studied
79in recent years, both computationally and theoretically.
80The original conjectures of Birch and
81Swinnerton-Dyer~\cite{birch-swd-I,birch-swd-II},
82for elliptic curves over $\Q$, were greatly influenced by
83computations.
84The scale of these computations was extended and systematized
85by Cremona in~\cite{cremona:algs}.
86
87In another direction, Wiles~\cite{wiles:fermat} and
88Taylor-Wiles~\cite{taylor-wiles} proved a special case of the
89conjecture of Shimura-Taniyama, which asserts that every elliptic
90curve over~$\Q$ is a quotient of some $J_0(N)$; this allowed them
91to establish Fermat's Last Theorem.  The full Shimura-Taniyama
92conjecture was later proved by Breuil, Conrad, Diamond, and Taylor
94This illustrates the central role played by quotients of $J_0(N)$.
95
96\section{Component Groups of $A_f$}
97
98The N\'eron model $\cA/\Z$ of an abelian variety $A/\Q$ is by
99definition a smooth commutative group scheme over~$\Z$ with
100generic fiber~$A$ such that for any smooth scheme~$S$ over~$\Z$,
101the restriction map $$\Hom_\Z(S,\cA) \rta \Hom_\Q(S_\Q,A)$$ is
102a bijection.  For more details, including a proof of existence,
103see, e.g.,~\cite{neronmodels}.
104
105Suppose that $A_f$ is a quotient of $J_0(N)$ corresponding to a
106newform~$f$ on $\Gamma_0(N)$,
107and let $\cA_f$ be the N\'eron model of~$A_f$.
108For any prime divisor~$p$ of~$N$, the closed fiber~${\cA_f}_{/\Fp}$
109is a group scheme over~$\Fp$, which need not be connected.
110Denote the connected component of the identity
111by~${\cA^{\circ}_f}_{/\Fp}$. There is an exact sequence
112$$1130 \rightarrow {\cA^{\circ}_f}_{/\Fp} 114 \rightarrow {\cA_f}_{/\Fp}\rightarrow\Phi_{A_f,p} 115 \rightarrow 0 116$$
117with $\Phi_{A_f,p}$ a finite \'{e}tale group scheme over~$\Fp$
118called the {\em component group} of $A_f$ at~$p$.
119
120The category of
121finite \'etale group schemes over $\Fp$ is
122equivalent  to  the category of
123finite groups equipped with an action of
124$\Gal(\Fpbar/\Fp)$ (see, e.g., \cite[\S6.4]{waterhouse}).
125The {\em order} of an \'etale group scheme $G/\Fp$ is defined
126to be the order of the group $G(\Fpbar)$.
127In this paper we describe an algorithm for computing the order of
128$\Phi_{A_f,p}$, when~$p$ exactly divides~$N$.
129
130\section{The Algorithm}
131
132Let~$J = J_0(N)$, fix a newform~$f$ of weight-two for $\Gamma_0(N)$,
133and let $A_f$ be the corresponding quotient of~$J$.  Because~$J$ is
134the Jacobian of a curve, it is canonically isomorphic to its dual, so
135the projection $J \rta A_f$ induces a polarization $A_f^{\vee} \rta 136A_f$, where $A_f^{\vee}$ denotes the abelian variety dual of $A_f$.
137We define the {\em modular degree} $\delta_{A_f}$ of $A_f$ to be the
138positive square root of the degree of this polarization.  This agrees
139with the usual notion of modular degree when $A_f$ is an elliptic curve.
140
141A {\it torus} $T$ over a field $k$ is a group scheme whose base
142extension to the separable closure $k_s$ of $k$ is a finite
143product of copies of $\G_m$.  Every commutative algebraic group
144over~$k$ admits a unique maximal subtorus, defined over~$k$,
145whose formation commutes with base extension (see IX \S2.1
146of~\cite{groth:sga7}).  The {\it character group}
147of a torus~$T$ is the group $\cX = \Hom_{k_s}(T,\G_m)$ which is
148a free abelian group of finite rank together with an action of
149$\Gal(k_s/k)$ (see, e.g.,~\cite[\S7.3]{waterhouse}).
150
151We apply this construction to our setting as follows.
152The closed fiber of the N\'eron model of $J$ at~$p$ is a group
153scheme over $\Fp$, whose maximal torus we denote by $T_{J,p}$.
154We define $\cX_{J,p}$ to be the character group of $T_{J,p}$.
155Then $\cX_{J,p}$ is a free abelian group equipped with an
156action of both $\Gal(\Fpbar/\Fp)$ and the Hecke algebra~$\T$
157(see, e.g.,~\cite{ribet:modreps}).
158Moreover, there exists a bilinear pairing
159$$160\langle\,,\,\rangle : \cX_{J,p} \times \cX_{J,p} \rightarrow \Z 161$$
162called the {\em monodromy pairing} such that
163$$164\Phi_{J,p} \isom \coker(\cX_{J,p} \rightarrow \Hom(\cX_{J,p},\Z)). 165$$
166Let $\cX_{J,p}[I_f]$ be the intersection of all kernels $\ker(t)$
167for~$t$
168in $I_f$, and let
169$$170\alpha_f: \cX_{J,p} \rightarrow \Hom(\cX_{J,p}[I_f],\Z) 171$$
172be the map induced by the monodromy pairing.
173The following theorem of the second author~\cite{stein:phd}, provides the
174basis for the computation of orders of component groups.
175
176\begin{theorem}\label{thm:main}
177With the notation as above, we have the equality
178$$179\#\Phi_{A_f,p} 180 = \frac{\#\coker(\alpha_f) \cdot \delta_{A_f}} 181 {\# (\alpha_f(\cX_{J,p})/\alpha_f(\cX_{J,p}[I_f]))}\,. 182$$
183\end{theorem}
184
185\subsection{Computing the modular degree $\delta_{A,f}$}
186Using modular symbols (see, e.g.,~\cite{cremona:algs}),
187we first compute the homology group $H_1(X_0(N),\Q;\mbox{\rm cusps})$.
188Using lattice reduction, we compute the $\Z$-submodule
189$H_1(X_0(N),\Z;\mbox{\rm cusps})$
190generated by all Manin symbols $(c,d)$.  Then
191$H_1(X_0(N),\Z)$ is the
192{\em integer} kernel of the boundary map.
193
194The Hecke ring $\T$ acts on $H_1(X_0(N),\Z)$ and also on
195the linear dual $\Hom(H_1(X_0(N),\Z),\Z)$,
196where $t \in \T$ acts on $\varphi \in \Hom(H_1(X_0(N),\Z),\Z)$
197by $(t.\varphi)(x) = \varphi(tx)$.
198We have a natural restriction map
199$$200r_f:\Hom(H_1(X_0(N),\Z),\Z)[I_f] \rightarrow 201 \Hom(H_1(X_0(N),\Z)[I_f],\Z). 202$$
203\begin{proposition}
204The cokernel of $r_f$ is isomorphic to the kernel
205of the polarization $A_f^{\vee}\rightarrow A_f$
206induced by the map $J_0(N)\rightarrow A_f$.
207\end{proposition}
208
209Thus the order of the cokernel of $r_f$ is the square of the modular
210degree~$\delta_f$.  We pause to note that the degree of any
211polarization is a square; see, e.g.,~\cite[Thm.~13.3]{milne:abvar}.
212\begin{proof}
213Let $S = S_2(\Gamma_0(N),\C)$ be the complex vector space of
214weight-two modular forms of level~$N$, and set $H = H_1(X_0(N),Z)$.
215The integration pairing $S\times H \rightarrow \C$
216induces a natural map
217$$\Phi_f:H\rightarrow \Hom(S[I_f],\C).$$
218Using the classical
219Abel-Jacobi theorem, we deduce the following commutative diagram,
220which has exact columns, but whose rows are not exact.
221$$\[email protected]=.5cm{ 222 0\ar[d] & 0\ar[d] & 0\ar[d] \\ 223 H[I_f]\ar[d]\ar[r] & H\ar[d]\ar[r]&\Phi_f(H)\ar[d] \\ 224\Hom(S,\C)[I_f]\ar[d]\ar[r] &\Hom(S,\C)\ar[d]\ar[r] &\Hom(S[I_f],\C)\ar[d]\\ 225 {A_f^{\vee}(\C)}\ar[d]\ar[r] 226 \[email protected]/_3.5pc/[rr]& J_0(N)(\C)\ar[d]\ar[r]& A_f(\C)\ar[d]\\ 227 0 & 0 & 0 \\ 228}$$
229By the snake lemma,
230the kernel of $A_f^{\vee}(\C)\rightarrow A_f(\C)$ is isomorphic to the
231cokernel of the map $H[I_f] \rightarrow \Phi_f(H)$.
232Since
233    $$\Hom(H/\ker(\Phi_f),\Z) \isom \Hom(H,\Z)[I_f],$$
234the $\Hom(-,\Z)$ dual of the map $H[I_f] \rightarrow \Phi_f(H)=H/\ker(\Phi_f)$
235is $r_f$, which proves the proposition.
236\end{proof}
237
238\subsection{Computing the character group $\cX_{J,p}$}
239Let $N = Mp$, where $M$ and $p$ are coprime.  If~$M$ is small,
240then the algorithm of Mestre and Oesterl\'e~\cite{mestre:graphs}
241can be used to compute $\cX_{J,p}$.  This algorithm constructs
242the graph of isogenies between $\Fpbar$-isomorphism classes of
243pairs consisting of a supersingular elliptic curve and a cyclic
244$M$-torsion subgroup.  In particular, the method is elementary
245to apply when $X_0(M)$ has genus~$0$.
246
247In general, the above category of enhanced'' supersingular
248elliptic curves can be replaced by one of left (or right) ideals
249of a quaternion order~$\cO$ of level~$M$ in the quaternion
250algebra over~$\Q$ ramified at~$p$.
251This gives an equivalent category, in which the computation of
252homomorphisms is efficient.  The character group $\cX_{J,p}$ is
253known by Deligne-Rapoport~\cite{deligne-rapoport} to be canonically
254isomorphic to the degree zero subgroup $\cX(\cO)$ of the free
255abelian divisor group'' on the isomorphism classes of enhanced
256supersingular  elliptic curves and of quaternion ideals.
257Moreover, this isomorphism  is compatible with the operation of
258Hecke operators, which are  effectively computable in $\cX(\cO)$
259in terms of ideal homomorphisms.
260
261The inner product of two classes in this setting is defined
262to be the number of isomorphisms between any two representatives.
263The linear extension to $\cX(\cO)$ gives an inner product which
264agrees, under the isomorphism, with the monodromy pairing on
265$\cX_{J,p}$.  This gives, in particular, an isomorphism $\Phi_{J,p} 266\isom \coker(\cX(\cO) \rightarrow \Hom(\cX(\cO),\Z))$, and an
267effective means of computing $\#\coker(\alpha_f)$ and
268$\#(\alpha_f(\cX_{J,p})/\alpha_f(\cX_{J,p}[I_f]))$.
269
270The arithmetic of quaternions has been implemented in
271{\sc Magma}~\cite{magma} by the first author.  Additional details
272and the application to Shimura curves, generalizing $X_0(N)$,
273can be found in Kohel~\cite{kohel}.
274
275\subsection{The Galois action on $\Phi_{A_f,p}$}
276
277To determine the Galois action on $\Phi_{A_f,p}$, we need only
278know the action of the Frobenius automorphism $\Frob_p$.
279However, $\Frob_p$ acts on $\Phi_{A_f,p}$ in the same way
280as $-W_p$, where $W_p$ is the
281$p$th Atkin-Lehner involution, which can be computed using modular
282symbols.  Since~$f$ is an eigenform, the involution $W_p$ acts
283as either $+1$ or $-1$ on $\Phi_{A_f,p}$.
284Moreover, the operator $W_p$ is determined by an involution on
285the set of quaternion ideals, so it can be determined explicitly
286on the character group.
287
288\section{Tables}
289
290The main computational results of this work are presented below
291in two tables.  The relevant algorithms have been implemented in
292{\sc Magma} and will be made part of a future release.
293They can also be obtained from the second author.
294
295\subsection{Component groups at low level}
296
297Table~\ref{tbl:lowlevel} gives the component groups of the
298quotients $A_f$ of $J_0(N)$ for $N\leq 127$.
299The column labeled $d$ contains the
300dimensions of the $A_f$,
301and the column labeled $\#\Phi_{A_f,p}$ contains a list
302of the orders of the component groups of $A_f$,
303one for each divisor~$p$ of~$N$, ordered by increasing~$p$.
304An entry of ?'' indicates that $p^2\mid N$, so our algorithm
305does not apply.
306A component group order is starred if the $\Gal(\Fpbar/\Fp)$-action is
307nontrivial.  More data along these lines can be obtained from
308the second author.
309
310\begin{table}
311\begin{center}
312\caption{Component groups at low level}
313\end{center}
314\label{tbl:lowlevel}
315$$316\begin{array}{lcl} 317 N & \, d \, & \, \#\Phi_{A_f,p}\, \\ 31811 & 1 & 5\\ 31914 & 1 & 6^*,3\\ 32015 & 1 & 4^*,4\\ 32117 & 1 & 4\\ 32219 & 1 & 3\\ 32320 & 1 & ?,2^*\\ 32421 & 1 & 4,2^*\\ 32523 & 2 & 11\\ 32624 & 1 & ?,2^*\\ 32726 & 1 & 3^*,3\\ 328 & 1 & 7,1^*\\ 32927 & 1 & ?\\ 33029 & 2 & 7\\ 33130 & 1 & 4^*,3,1^*\\ 33231 & 2 & 5\\ 33332 & 1 & ?\\ 33433 & 1 & 6^*,2\\ 33534 & 1 & 6,1^*\\ 33635 & 1 & 3^*,3\\ 337 & 2 & 8,4^*\\ 33836 & 1 & ?,?\\ 33937 & 1 & 1^*\\ 340 & 1 & 3\\ 34138 & 1 & 9^*,3\\ 342 & 1 & 5,1^*\\ 34339 & 1 & 2^*,2\\ 344 & 2 & 14,2^*\\ 34540 & 1 & ?,2\\ 34641 & 3 & 10\\ 34742 & 1 & 8,2^*,1^*\\ 34843 & 1 & 1^*\\ 349 & 2 & 7\\ 35044 & 1 & ?,1^*\\ 35145 & 1 & ?,1^*\\ 35246 & 1 & 10^*,1\\ 35347 & 4 & 23\\ 35448 & 1 & ?,2\\ 35549 & 1 & ?\\ 35650 & 1 & 1^*,?\\ 357 & 1 & 5,?\\ 35851 & 1 & 3,1^*\\ 359 & 2 & 16^*,4\\ 36052 & 1 & ?,2^*\\ 36153 & 1 & 1^*\\ 362\end{array}\quad 363\begin{array}{lcl} 364 N & \, d \, & \, \#\Phi_{A_f,p}\, \\ 365 & 3 & 13\\ 36654 & 1 & 3^*,?\\ 367 & 1 & 3,?\\ 36855 & 1 & 2,2^*\\ 369 & 2 & 14^*,2\\ 37056 & 1 & ?,1\\ 371 & 1 & ?,1^*\\ 37257 & 1 & 2^*,1^*\\ 373 & 1 & 2,2^*\\ 374 & 1 & 10,1^*\\ 37558 & 1 & 2^*,1^*\\ 376 & 1 & 10,1^*\\ 37759 & 5 & 29\\ 37861 & 1 & 1^*\\ 379 & 3 & 5\\ 38062 & 1 & 4,1^*\\ 381 & 2 & 66^*,3\\ 38263 & 1 & ?,1^*\\ 383 & 2 & ?,3\\ 38464 & 1 & ?\\ 38565 & 1 & 1^*,1^*\\ 386 & 2 & 3^*,3\\ 387 & 2 & 7,1^*\\ 38866 & 1 & 2^*,3,1^*\\ 389 & 1 & 4,1^*,1^*\\ 390 & 1 & 10,5,1\\ 39167 & 1 & 1\\ 392 & 2 & 1^*\\ 393 & 2 & 11\\ 39468 & 2 & ?,2^*\\ 39569 & 1 & 2,1^*\\ 396 & 2 & 22^*,2\\ 39770 & 1 & 4,2^*,1^*\\ 39871 & 3 & 5\\ 399 & 3 & 7\\ 40072 & 1 & ?,?\\ 40173 & 1 & 2\\ 402 & 2 & 1^*\\ 403 & 2 & 3\\ 40474 & 2 & 9^*,3\\ 405 & 2 & 95,1^*\\ 40675 & 1 & 1^*,?\\ 407 & 1 & 1,?\\ 408 & 1 & 5,?\\ 409\end{array}\quad 410\begin{array}{lcl} 411 N & \, d \, & \, \#\Phi_{A_f,p}\, \\ 41276 & 1 & ?,1^*\\ 41377 & 1 & 2^*,1^*\\ 414 & 1 & 3^*,2\\ 415 & 1 & 6,3^*\\ 416 & 2 & 2,2^*\\ 41778 & 1 & 16^*,5^*,1\\ 41879 & 1 & 1^*\\ 419 & 5 & 13\\ 42080 & 1 & ?,2\\ 421 & 1 & ?,2^*\\ 42281 & 2 & ?\\ 42382 & 1 & 2^*,1^*\\ 424 & 2 & 28,1^*\\ 42583 & 1 & 1^*\\ 426 & 6 & 41\\ 42784 & 1 & ?,1^*,2^*\\ 428 & 1 & ?,3,2\\ 42985 & 1 & 2^*,1\\ 430 & 2 & 2^*,1^*\\ 431 & 2 & 6,1^*\\ 43286 & 2 & 21^*,3\\ 433 & 2 & 55,1^*\\ 43487 & 2 & 5,1^*\\ 435 & 3 & 92^*,4\\ 43688 & 1 & ?,1^*\\ 437 & 2 & ?,2^*\\ 43889 & 1 & 1^*\\ 439 & 1 & 2\\ 440 & 5 & 11\\ 44190 & 1 & 2^*,?,3\\ 442 & 1 & 6,?,1^*\\ 443 & 1 & 4,?,1\\ 44491 & 1 & 1^*,1^*\\ 445 & 1 & 1,1\\ 446 & 2 & 7,1^*\\ 447 & 3 & 4^*,8\\ 44892 & 1 & ?,1^*\\ 449 & 1 & ?,1\\ 45093 & 2 & 4^*,1^*\\ 451 & 3 & 64,2^*\\ 45294 & 1 & 2,1^*\\ 453 & 2 & 94^*,1\\ 45495 & 3 & 10,2^*\\ 455 & 4 & 54^*,6\\ 456\end{array}\quad 457\begin{array}{lcl} 458 N & \, d \, & \, \#\Phi_{A_f,p}\, \\ 45996 & 1 & ?,2\\ 460 & 1 & ?,2^*\\ 46197 & 3 & 1^*\\ 462 & 4 & 8\\ 46398 & 1 & 2^*,?\\ 464 & 2 & 14,?\\ 46599 & 1 & ?,1^*\\ 466 & 1 & ?,1\\ 467 & 1 & ?,1^*\\ 468 & 1 & ?,1^*\\ 469100 & 1 & ?,?\\ 470101 & 1 & 1^*\\ 471 & 7 & 25\\ 472102 & 1 & 2^*,2^*,1^*\\ 473 & 1 & 6^*,6,1^*\\ 474 & 1 & 8,4,1\\ 475103 & 2 & 1^*\\ 476 & 6 & 17\\ 477104 & 1 & ?,1^*\\ 478 & 2 & ?,2\\ 479105 & 1 & 1,1,1\\ 480 & 2 & 10^*,2^*,2\\ 481106 & 1 & 4^*,1^*\\ 482 & 1 & 5^*,1\\ 483 & 1 & 24,1^*\\ 484 & 1 & 3,1^*\\ 485107 & 2 & 1^*\\ 486 & 7 & 53\\ 487108 & 1 & ?,?\\ 488109 & 1 & 1\\ 489 & 3 & 1^*\\ 490 & 4 & 9\\ 491110 & 1 & 7^*,1^*,3\\ 492 & 1 & 3,1^*,1^*\\ 493 & 1 & 5,5,1\\ 494 & 2 & 16^*,3,1^*\\ 495111 & 3 & 10^*,2\\ 496 & 4 & 266,2^*\\ 497112 & 1 & ?,1^*\\ 498 & 1 & ?,1\\ 499 & 1 & ?,1^*\\ 500113 & 1 & 2\\ 501 & 2 & 2\\ 502 & 3 & 1^*\\ 503\end{array}\quad 504\begin{array}{lcl} 505 N & \, d \, & \, \#\Phi_{A_f,p}\, \\ 506 & 3 & 7\\ 507114 & 1 & 2^*,5^*,1\\ 508 & 1 & 20,3^*,1^*\\ 509 & 1 & 6,3,1\\ 510115 & 1 & 5^*,1\\ 511 & 2 & 4^*,1^*\\ 512 & 4 & 32,4^*\\ 513116 & 1 & ?,1^*\\ 514 & 1 & ?,2^*\\ 515 & 1 & ?,1^*\\ 516117 & 1 & ?,1\\ 517 & 2 & ?,3\\ 518 & 2 & ?,1^*\\ 519118 & 1 & 2^*,1^*\\ 520 & 1 & 19^*,1\\ 521 & 1 & 10,1^*\\ 522 & 1 & 1,1^*\\ 523119 & 4 & 9,3^*\\ 524 & 5 & 48^*,8\\ 525120 & 1 & ?,1,1^*\\ 526 & 1 & ?,2,1\\ 527121 & 1 & ?\\ 528 & 1 & ?\\ 529 & 1 & ?\\ 530 & 1 & ?\\ 531122 & 1 & 4^*,1^*\\ 532 & 2 & 39^*,3\\ 533 & 3 & 248,1^*\\ 534123 & 1 & 1^*,1^*\\ 535 & 1 & 5,1\\ 536 & 2 & 7,1^*\\ 537 & 3 & 184^*,4\\ 538124 & 1 & ?,1^*\\ 539 & 1 & ?,1\\ 540125 & 2 & ?\\ 541 & 2 & ?\\ 542 & 4 & ?\\ 543126 & 1 & 8^*,?,1^*\\ 544 & 1 & 2,?,1\\ 545127 & 3 & 1^*\\ 546 & 7 & 21\\ 547&&\\ 548&&\\ 549&&\\ 550\end{array}$$
551\end{table}
552
553
554\subsection{Examples of large component groups}
555
556Let $\Omega_{A_f}$ be the real period of $A_f$, as defined by
557J.~Tate in~\cite{tate:bsd}.   The second author computed
558the rational numbers $L(A_f,1)/\Omega_{A_f}$ for every
559newform~$f$ of level $N\leq 1500$.
560The five largest prime divisors occur in the ratios
561given in Table~\ref{table:lratios}.
562The Birch and Swinnerton-Dyer conjecture predicts that the large
563prime divisor of the numerator of each special value must
564divide the order either of some component group $\Phi_{A_f,p}$ or of the
565Shafarevich-Tate group of~$A_f$.  In each instance
566$\Phi_{A_f,2}$ is divisible by the large prime divisor, as
567predicted.
568
569\begin{table}
570\label{table:lratios}
571\begin{center}
572\caption{Large $L(A_f,1)/\Omega_{A_f}$}
573\end{center}
574$$\begin{array}{ccll} 575 \qquad N \qquad\quad & 576 \quad \dim \quad & 577 \quad\quad L(A_f,1)/\Omega_{A_f} \qquad\qquad & 578 \qquad \#\Phi_{A_f,p} \\ 579 1154=2\tdot 577 & 20 & 2^?\tdot85495047371/17^2 580 & 2^?\tdot 17^2 \tdot 85495047371, 2^? \\ 581 1238=2\tdot 619 & 19 & 2^?\tdot 7553329019/5\tdot 31 582 & 2^?\tdot 5\tdot31\tdot7553329019 , 2^?\\ 583 1322=2\tdot 661 & 21 & 2^?\tdot 57851840099/331 584 & 2^?\tdot 331 \tdot 57851840099, 2^?\\ 585 1382=2\tdot 691 & 20 & 2^?\tdot 37 \tdot 1864449649 /173 586 & 2^?\tdot 37\tdot173\tdot 1864449649, 2^?\\ 587 1478=2\tdot 739 & 20 & 588 2^?\tdot 7\tdot 29\tdot 1183045463 / 5\tdot 37 589 & 2^?\tdot5\tdot 7\tdot29\tdot37\tdot1183045463, 2^?\\ 590\end{array}$$
591\end{table}
592
593\section{Further directions}
594
595Further considerations are needed to compute the {\em group}
596structure of $\Phi_{A_f,p}$.  However, since the action of Frobenius
597is known, computing the group structure of $\Phi_{A_f,p}$ suffices
598to determine its structure as a group scheme.
599
600%An equivalence with quaternion divisor groups is not known to
601%hold for the character group at~$p$ when $p^2$ divides~$N$.
602%Thus
603Our methods say nothing about the component group at primes
604whose {\em square} divides the level.  The free abelian group
605on classes of nonmaximal orders of index~$p$ at a ramified prime
606gives a well-defined divisor group.
607Do the resulting Hecke modules determine the component groups
608for quotients of level $p^2M$?
609
610Is it possible to define quantities as in Theorem~\ref{thm:main}
611even when the weight of~$f$ is {\em greater than~$2$}?
612If so, how are the resulting quantities related to the Bloch-Kato
613Tamagawa  numbers (see~\cite{bloch-kato}) of the higher weight
614motive attached to~$f$?
615
616\newpage
617\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
618\begin{thebibliography}{1}
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724\end{thebibliography}
725
726\end{document}
727
728
729Oh, and the tables now get pushed into the middle of
730the references.  If you have a quick and dirty way
731of fixing this, then please do so as I find it ugly.
732Hmmm, I played slightly with this to no success, but
733also noted that I get no table numbers, at least as
734it compiles on the system here.  Could you look at
735this?  Sorry for leaving you this dirty work.
736
737-----------------------------------------------------------------
738
739Typographical errors or syntax:
740
741- page 1:  "-" between One and dimensional?
742- Sec 2 line 1/2: mistaken \par inserted?
743- Sec 3.1 before display: mistaken \par inserted?
744- sentence on pagebreak 3/4: do you mean canonically isomorphic
745  instead of canonically equivalent?  That would help to interpret the
746  next sentence.
747- page 4 isomophism (maybe spell check the whole document?)
748- Sec 3.3 last sentence: "so can" --> "so it can"
749- Sec 4.2 "either the order of" --> "the order of either" ?
750
751Corrected.
752
754
755- last line of sec 2:
756  how can two objects in distinct categories be equivalent?
757  Maybe say that GIVING one is equivalent to GIVING the other.
758
759We correct this by saying that the (implicit) parent categories
760are equivalent rather than two objects in them.
761
762- Section 3: can you explain or give a reference for what the
763  "toric part" is.  The average ANTS reader would appreciate it.
764  It is not a subgroup or a quotient, but a subquotient, right?
765
766We add a short paragraph to define a torus and the existence
767of a maximal torus, with references.
768
769- Sec 3.1 last sentence: You view abelian varieties as
770  complex lattices?  That sounds like a stretch--or do you have
771  some equivalence in mind.  Maybe view their fundamental groups
772  as complex lattices?
773
774We rewrote this section for clarity, and added a proof of the
775formula for the modular degree.
776
777-----------------------------------------------------------------
778